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A stabilized finite element method based on two local Gauss integrations for the Stokes equations. (English) Zbl 1132.35436
Summary: This paper considers a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for the Stokes equations approximated by the lowest equal-order finite element pairs. This method offsets only the discrete pressure space by subtracting the simple and symmetrical term at element level in order to circumvent the inf-sup condition. Optimal error estimates are obtained by applying the standard Galerkin technique. Finally, the numerical illustrations are presented completely in agree with the theoretical expectations.

MSC:
35Q30 Navier-Stokes equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Barth, T.; Bochev, P.; Gunzburger, M.D.; Shadid, J.N., A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. sci. comput., 25, 1585-1607, (2004) · Zbl 1133.76307
[2] Becker, R.; Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 4, 211-224, (2001) · Zbl 1008.76036
[3] Bochev, P.B.; Dohrmann, C.R.; Gunzburger, M.D., Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. numer. anal., 44, 82-101, (2006) · Zbl 1145.76015
[4] Bochev, P.B.; Gunzburger, M.D., An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations, SIAM J. numer. anal., 42, 1189-1207, (2004) · Zbl 1159.76346
[5] Brefort, B.; Ghidaglia, J.M.; Temam, R., Attractor for the penalty navier – stokes equations, SIAM J. math. anal., 19, 1-21, (1988) · Zbl 0696.35131
[6] Brezzi, F.; Douglas, J., Stabilized mixed methods for the Stokes problem, Numer. math., 53, 225-235, (1988) · Zbl 0669.76052
[7] Brezzi, F.; Fortin, M., A minimal stabilisation procedure for mixed finite element methods, Numer. math., 3, 457-491, (2001) · Zbl 1009.65067
[8] E. Burman, A note on pressure projection stabilizations for Galerkin approximations of Stokes problem, Numer. Meth. Part D E, to appear. · Zbl 1139.76029
[9] Buscagliaa, G.C.; Basombrio, F.G.; Codinab, R., Fourier analysis of an equal-order incompressible flow solver stabilized by pressure gradient projection, Internat. J. numer. methods fluids, 34, 65-92, (2000) · Zbl 0985.76049
[10] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[11] Dohrmann, C.R.; Bochev, P., A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Internat. J. numer. methods fluids, 46, 183-201, (2004) · Zbl 1060.76569
[12] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comput., 52, 495-508, (1989) · Zbl 0669.76051
[13] Gerdes, K.; Schöktza, D., Hp-finite element simulations for Stokes flow— stable and stabilized, Finite elements anal. design, 33, 143-165, (1999) · Zbl 0965.76043
[14] Girault, V.; Raviart, P.A., Finite element method for navier – stokes equations: theory and algorithms, (1987), Springer Berlin, Heidelberg · Zbl 0396.65070
[15] Hughes, T.; Franca, L.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska – brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. methods appl. mech. eng., 59, 85-99, (1986) · Zbl 0622.76077
[16] Layton, W., Model reduction by constraints, discretization of flow problems and an induced pressure stabilization, Numer. linear algebra appl., 12, 547-562, (2005) · Zbl 1164.76364
[17] Layton, W.; Tobiska, L., A two-level method with backtraking for the navier – stokes equations, SIAM J. numer. anal., 35, 2035-2054, (1998) · Zbl 0913.76050
[18] Silvester, D.J., Optimal low-order finite element methods for incompressible flow, Comput. methods appl. mech. eng., 111, 357-368, (1994) · Zbl 0844.76059
[19] D.J. Silvester, Stabilised mixed finite element methods, Numerical Analysis Report No. 262, 1995.
[20] D.J. Silvester, Stabilised vs stable mixed finite element methods for incompressible flow, Numerical Analysis Report No. 307, 1997.
[21] Silvester, D.J.; Kechkar, N., Stabilized bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. methods appl. mech. eng., 79, 71-86, (1990) · Zbl 0706.76075
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